Problem: Solve for $x$ : $ 4|x - 9| + 7 = 6|x - 9| + 8 $
Subtract $ {4|x - 9|} $ from both sides: $ \begin{eqnarray} 4|x - 9| + 7 &=& 6|x - 9| + 8 \\ \\ {- 4|x - 9|} && {- 4|x - 9|} \\ \\ 7 &=& 2|x - 9| + 8 \end{eqnarray} $ Subtract $8$ from both sides: $ \begin{eqnarray} 7 &=& 2|x - 9| + 8 \\ \\ {- 8} && {- 8} \\ \\ -1 &=& 2|x - 9| \end{eqnarray} $ Divide both sides by ${2}$ $ \dfrac{-1} {{2}} = \dfrac{2|x - 9|} {{2}} $ Simplify: $ -\dfrac{1}{2} = |x - 9| $ The absolute value cannot be negative. Therefore, there is no solution.